[Paper Review] Optimal Trade-offs in Multi-Processor Approximate Message Passing

Junan Zhu, Dror Baron+1 more|arXiv (Cornell University)|Jan 15, 2016

Sparse and Compressive Sensing Techniques32 references3 citations

TL;DR

This paper proposes an optimal coding rate allocation strategy for multi-processor approximate message passing (MP-AMP) systems using dynamic programming to minimize combined computation and communication costs while achieving a target estimation quality. It proves that in the low excess mean squared error (EMSE) regime, optimal coding rates grow approximately linearly per iteration, and the total cost scales as $O(\log^2(1/\text{EMSE}))$, enabling efficient trade-offs between performance, latency, and bandwidth in distributed linear inverse problems.

ABSTRACT

We consider large-scale linear inverse problems in Bayesian settings. We follow a recent line of work that applies the approximate message passing (AMP) framework to multi-processor (MP) computational systems, where each processor node stores and processes a subset of rows of the measurement matrix along with corresponding measurements. In each MP-AMP iteration, nodes of the MP system and its fusion center exchange lossily compressed messages pertaining to their estimates of the input. In this setup, we derive the optimal per-iteration coding rates using dynamic programming. We analyze the excess mean squared error (EMSE) beyond the minimum mean squared error (MMSE), and prove that, in the limit of low EMSE, the optimal coding rates increase approximately linearly per iteration. Additionally, we obtain that the combined cost of computation and communication scales with the desired estimation quality according to $O(\log^2(1/ ext{EMSE}))$. Finally, we study trade-offs between the physical costs of the estimation process including computation time, communication loads, and the estimation quality as a multi-objective optimization problem, and characterize the properties of the Pareto optimal surfaces.

Motivation & Objective

• To minimize the combined cost of computation and communication in multi-processor approximate message passing (MP-AMP) systems while achieving a desired estimation quality.
• To derive optimal per-iteration coding rates for lossy compression of messages exchanged between processor nodes and a fusion center in MP-AMP.
• To characterize the trade-offs between estimation quality (EMSE), computation time, and communication load as a multi-objective optimization problem.
• To establish theoretical scaling laws for the total cost as a function of desired estimation accuracy (EMSE) in the low-EMSE regime.

Proposed method

• Uses dynamic programming to compute the optimal sequence of coding rates that minimize total cost (computation + communication) for a given target EMSE.
• Applies rate-distortion theory principles to model lossy compression of messages exchanged in MP-AMP, accounting for error propagation across iterations.
• Derives a convex optimization formulation for the low-EMSE regime, enabling analytical characterization of optimal coding rate evolution.
• Employs state evolution formalism to model the mean squared error dynamics of MP-AMP and quantify the excess mean squared error (EMSE) above the minimum possible (MMSE).
• Uses Taylor series approximations and asymptotic analysis to derive the linear growth behavior of optimal coding rates in the low-EMSE limit.
• Characterizes the Pareto optimal surface for the multi-objective optimization problem involving estimation quality, computation, and communication.

Experimental results

Research questions

• RQ1 What is the optimal sequence of coding rates across iterations in MP-AMP that minimizes the combined cost of computation and communication for a given target estimation quality?
• RQ2 How do the optimal coding rates scale with the desired estimation accuracy (EMSE) in the low-EMSE regime?
• RQ3 What is the theoretical scaling of the total cost (computation + communication) as a function of the desired EMSE in MP-AMP systems?
• RQ4 How do the estimation error (EMSE) and communication load interact across iterations in a distributed MP-AMP system?
• RQ5 What is the structure of the Pareto optimal surface in the multi-objective trade-off between computation time, communication load, and estimation quality?

Key findings

• The optimal coding rate sequence grows approximately linearly with the iteration number in the low-EMSE regime, with a theoretical growth rate of $\frac{1}{2}\log_2\left(\frac{1}{\theta}\right)$ per iteration.
• The combined cost of computation and communication scales as $O(\log^2(1/\text{EMSE}))$ in the low-EMSE limit, which is significantly more efficient than linear scaling.
• The optimal coding rate at each iteration is derived via dynamic programming and proven to be globally optimal for minimizing total cost under a fixed EMSE constraint.
• In the low-EMSE regime, the ratio of optimal distortion levels between consecutive iterations converges to $\theta$, where $\theta$ is the decay factor of the state evolution variance.
• The optimal EMSE at iteration $t$ is $\epsilon_t^* = \frac{M}{N}\widetilde{u}_t^*$, and the ratio $\frac{D_t^*}{\widetilde{u}_t^*}$ tends to zero as $t \to \infty$, indicating diminishing relative distortion growth.
• The analysis confirms that the optimal coding rate sequence is insensitive to the choice of compression quantization bounds $C = \pm B$, with the leading-order behavior bounded by the extremes.